Problem: Simplify and expand the following expression: $ \dfrac{2x}{5x + 5}-\dfrac{4x}{x - 7} $
Answer: In order to subtract expressions, they must have a common denominator. Get both fractions over a common denominator of $(5x + 5)(x - 7)$ Multiply the first term by $\dfrac{x - 7}{x - 7}$ $ \begin{align*} \dfrac{2x}{5x + 5} \times \dfrac{x - 7}{x - 7} & = \dfrac{(2x)(x - 7)}{(5x + 5)(x - 7)} \\ & = \dfrac{2x^2 - 14x}{(5x + 5)(x - 7)}\end{align*} $ Multiply the second term by $\dfrac{5x + 5}{5x + 5}$ $ \begin{align*} \dfrac{4x}{x - 7} \times \dfrac{5x + 5}{5x + 5} & = \dfrac{(4x)(5x + 5)}{(x - 7)(5x + 5)} \\ & = \dfrac{20x^2 + 20x}{(x - 7)(5x + 5)}\end{align*} $ Now we have: $ = \dfrac{2x^2 - 14x}{(5x + 5)(x - 7)} - \dfrac{20x^2 + 20x}{(x - 7)(5x + 5)} $ Now both terms have a common denominator we can subtract the numerators: $ = \dfrac{2x^2 - 14x - (20x^2 + 20x)}{(5x + 5)(x - 7)} $ $ = \dfrac{2x^2 - 14x - 20x^2 - 20x}{(5x + 5)(x - 7)} $ $ = \dfrac{-18x^2 - 34x}{(5x + 5)(x - 7)}$ Expand the denominator: $ = \dfrac{-18x^2 - 34x}{5x^2 - 30x - 35}$